${\cal O} (α^2)$ Next-to-Leading Photonic Corrections to Small-Angle Bhabha Scattering in the Structure Function Formalism

A general method for computing ${\cal O} (\alpha^2)$ and higher-order next-to-leading photonic corrections is presented and applied to the precision calculation of the small-angle Bhabha scattering cross section in the phase-space region of interest for the luminosity measurement at LEP. The formulation is based on a proper matching of exact $\cal O (\alpha)$ results with higher-order corrections in the Structure Function formalism. The results of the approach are analytically compared with theoretical calculations, both for $s$- and $t$-channel processes, available for simple Event Selections. Numerical predictions for realistic Event Selections are also provided and critically compared with the ones existing in the literature.